In this paper, we study the exact recovery problem in the Gaussian weighted version of the Stochastic block model with two symmetric communities. We provide the information-theoretic threshold in terms of the signal-to-noise ratio (SNR) of the model and prove that when SNR $<1$, no statistical estimator can exactly recover the community structure with probability bounded away from zero. On the other hand, we show that when SNR $>1$, the Maximum likelihood estimator itself succeeds in exactly recovering the community structure with probability approaching one. Then, we provide two algorithms for achieving exact recovery. The Semi-definite relaxation as well as the spectral relaxation of the Maximum likelihood estimator can recover the community structure down to the threshold value of 1, establishing the absence of an information-computation gap for this model. Next, we compare the problem of community detection with the problem of recovering a planted densely weighted community within a graph and prove that the exact recovery of two symmetric communities is a strictly easier problem than recovering a planted dense subgraph of size half the total number of nodes, by establishing that when the same SNR$< 3/2$, no statistical estimator can exactly recover the planted community with probability bounded away from zero. More precisely, when $1 <$ SNR $< 3/2$ exact recovery of community detection is possible, both statistically and algorithmically, but it is impossible to exactly recover the planted community, even statistically, in the Gaussian weighted model. Finally, we show that when SNR $>2$, the Maximum likelihood estimator itself succeeds in exactly recovering the planted community with probability approaching one. We also prove that the Semi-definite relaxation of the Maximum likelihood estimator can recover the planted community structure down to the threshold value of 2.

翻译：本文研究了具有两个对称社区的随机块模型在高斯加权版本中的精确恢复问题。我们根据模型的信噪比（SNR）给出了信息论阈值，并证明当SNR < 1时，没有任何统计估计器能够以非零概率精确恢复社区结构。另一方面，我们证明当SNR > 1时，最大似然估计器本身能够以趋近于1的概率成功精确恢复社区结构。随后，我们提出了两种实现精确恢复的算法：最大似然估计器的半定松弛和谱松弛均能在阈值1以下恢复社区结构，从而证明了该模型不存在信息-计算鸿沟。接着，我们将社区检测问题与恢复图中种植的稠密加权社区问题进行比较，并通过证明当相同SNR < 3/2时，没有任何统计估计器能够以非零概率精确恢复种植社区，确立了恢复两个对称社区比恢复大小为总节点数一半的种植稠密子图更为简单。具体而言，当1 < SNR < 3/2时，社区检测在统计和算法上均可实现精确恢复，但在高斯加权模型中，即使从统计角度也无法精确恢复种植社区。最后，我们证明当SNR > 2时，最大似然估计器本身能够以趋近于1的概率成功精确恢复种植社区，并证明最大似然估计器的半定松弛可在阈值2以下恢复种植社区结构。